I don't think you need a map. You don't need to plan out movement across the streets. General information is all that's required. When the PCs decide to go to the Blue Bugbear, they only need to know that it's in the Tower Quarter, and they need to cross Dead Man's Bridge to get there. They don't need to know each street they take and every turn they make. All the information you get from a map is unneccesary.
When you draw your "map", just put down the important buildings. Here's the castle, here are the gates, here's the bridge, here's the market, etc. You can fill things in as time goes by (here's the sewer entrance, here's where Guard Captain Chuck - who hates you all - lives, here's the secret entrance, etc.).
